The three components of the bar and hinge model are shown in a way that can be used to analyze the foldable origami tube.

Analytical models for thin-sheet structures
We have improved a model for the analysis of origami structures that consist of flat panels interconnected by a pattern of prescribed fold lines. This bar and hinge model, first introduced by Schenk and Guest (2011)
can capture the fundamental behaviors of thin folded sheet structures. The model consists of three components: (1) elastic bar elements that simulate the stretching and shear stiffness thin panels; (2) rotational hinges
that simulate folding of the panels; and (3) rotational hinges that simulate folding along the more flexible prescribed fold lines. We have made the model scalable, and have incorporated material characteristics such
as the elastic modulus, Poisson's ratio, and thickness of the thin sheet. Although this model is not as as accurate as a detailed finite element (FE) model it provides several useful advantages. The model is easy to
implement, easy to use, it provides insightful results, and it is faster than FE analyses. The efficiency of the model makes it suitable for extensions to parametric studies, optimization or various specialized analyses.

Deployment and retraction of zipper-coupled tubes by actuating only on the right end.

Deployable coupled tube structures
We have explored a variety of origami tube systems and assemblages. The basic type of tube is constructed by placing two symmetric Miura-ori sheets together (see Tachi 2009). The tube is rigid and flat foldable meaning it
can fully unfold from a flattened state with deformations occurring only at the fold lines. We used eigenvalue and structural cantilever analysis to investigate and compare different geometries of tubes and coupled tube
systems. The "zipper"-coupled tube system (shown on the left) yields an unusually large eigenvalue band-gap that represents a unique difference in stiffness between deformation modes. The structure
has only one flexible mode through which it can deploy, yet it is significantly stiffer for all other bending and twisting type modes. The deployment motion is permitted by the flexible bending the thin sheet along the
prescribed fold lines, however all other modes require the significantly stiffer stretching and shear of the thin sheet. The zipper-couped tubes have the advantages of deployable origami, but also the stiffening
effect that is common in cellular/corrugated structures and materials.

(Top) A cellular metamterial with the stiffness varying based on the structural configuration. (Bottom) Deployable bridge structure constructed from two different tubes.

Extensions from metamaterials to deployable architecture
Origami sheets and origami tubes can be coupled, combined, and arranged in a variety of methods to form new geometries and structures. We have shown different methods in which the zipper-coupled tubes can be
assembled into cellular assemblages. By combining different types of coupled tubes together we can also enhance the structural characteristics of these systems. For example, the cubic cellular assemblage
(shown top right) consists of zipper and aligned coupling, and has both space filling properties and the enhanced stiffness of the zipper tubes. This assemblage can have a variable asymmetrical stiffness
depending on its configuration. Similarly, it is be possible to couple different geometries of tubes. To create a bridge type structure (shown bottom right), we use nearly square tubes to provide
a smooth deck, and we use zippered zig-zag tubes to create a stiffer parapet. The deployable origami assemblages could lead to practical applications ranging in size from microscale metamaterials that harness
the novel mechanical properties to large-scale deployable systems in engineering and architecture.
Publications related to this research:

Multiresolution Topology Optimization
We use versatile polygonal elements along with a multiresolution scheme for topology optimization. This approach allows for a computationally efficient and high resolution design for structural
dynamics problems. The polygonal elements allow for fast and easy meshing of complex domains, or uneven refinements in discretization. The multiresolution scheme uses a coarse finite element
mesh to perform the analysis, a fine design variable mesh for the optimization and a fine density variable mesh to represent the material distribution. We show applications of this multiresolution approach
in the topology optimization of structural eigenfrequencies and forced vibration problems. With the multiresolution approach the finite element analysis on a coarse mesh is efficient, and a higher resolution can be obtained by
distributing material in a finer mesh of design/density variables (part d in figure).
Publications related to this research:

(Top) Lateral loading of an elastomeric bearing with side retainers (Bottom) Longitudinal deformation of the full bridge model used for seismic analyses.

Quasi-Isolation Systems for Bridges
We explore "quasi-isolation" which is a modern philosophy for seismic design of bridges. With this approach nonlinearity occurs in specific bearing components such that forces transferred into the
substructure are reduced and isolation is achieved by sliding of bearings. This system can provide a low-complexity, low-cost approach to mitigate of earthquake effects in locations with risk at long
recurrence periods, such as the eastern and central United States. The proposed system employs a set of fixed bearings at one intermediate substructure,
and all other substructures are instrumented with elastomeric bearings that permit thermal expansion (top left). L-shaped steel side retainers are placed in the transverse direction of the elastomeric bearings. Along with
the low-profile fixed bearings, these components prevent bridge movement during service loading, but break-off and permit sliding at high earthquake loads.
We construct a finite element model (bottom right) that can capture a variety of nonlinear behaviors in the bridge and in the bearing elements.
New models are formulated to capture the bi-directional stick-slip behavior in the sliding bearings and the bilinear (and eventual fracture) behavior of steel retainers and fixed bearings. These models are informed and
calibrated based on a detailed experimental study. Longitudinal and transverse pushover analyses are performed to demonstrate local limit states and progression of damage in the bridge structure.
A large scale parametric study with incremental dynamic analyses is carried out to investigate the quasi-isolated system on a variety of different bridges. Results indicate that bridges with quasi-isolation bearings can
be resilient even for high seismic events in Illinois. However, the approach can be refined to improve performance and reduce damage to components such as the intermediate bridge piers.
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